the thom isomorphism in gauge-equivariant k-theorymech.math.msu.su/~troitsky/mpim2005-47.pdf · the...

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THE THOM ISOMORPHISM IN GAUGE-EQUIVARIANT K-THEORY VICTOR NISTOR AND EVGENIJ TROITSKY Abstract. In a previous paper [15], we have introduced the gauge-equivariant K-theory group K 0 G (X) of a bundle π X : X B endowed with a continuous action of a bundle of compact Lie groups p : G→ B. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e., a family of elliptic operators invariant with respect to the ac- tion of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Bundles of compact groups and finite holonomy conditions 3 2.2. Gauge-equivariant K-theory 4 2.3. Additional results 6 3. K-theory and complexes 8 3.1. The L n G –groups 8 3.2. Euler characteristics 10 3.3. Globally defined complexes 14 3.4. The non-compact case 15 4. The Thom isomorphism 15 4.1. Products 16 4.2. The non-compact case 17 5. Gysin maps 20 6. The topological index 24 References 29 1. Introduction In this paper we establish a Thom isomorphism theorem for gauge equivariant K-theory. Let p : G→ B be a bundle of compact groups. Recall that this means that each fiber G b := p -1 (b) is a compact group and that, locally, G is of the form V. N. was partially supported by NSF Grants DMS-9971951, DMS 02-00808, and a “collabo- rative research” grant. E. T. was partially supported by RFFI Grant 05-01-00923, Grant for the support of leading scientific schools and Grant “Universities of Russia” YP.04.02.530. The present joint research was started under the hospitality of MPIM (Bonn). 1

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Page 1: THE THOM ISOMORPHISM IN GAUGE-EQUIVARIANT K-THEORYmech.math.msu.su/~troitsky/MPIM2005-47.pdf · THE THOM ISOMORPHISM IN GAUGE-EQUIVARIANT K-THEORY VICTOR NISTOR AND EVGENIJ TROITSKY

THE THOM ISOMORPHISM IN GAUGE-EQUIVARIANT

K-THEORY

VICTOR NISTOR AND EVGENIJ TROITSKY

Abstract. In a previous paper [15], we have introduced the gauge-equivariantK-theory group K0

G(X) of a bundle πX : X → B endowed with a continuous

action of a bundle of compact Lie groups p : G → B. These groups arethe natural range for the analytic index of a family of gauge-invariant ellipticoperators (i.e., a family of elliptic operators invariant with respect to the ac-tion of a bundle of compact groups). In this paper, we continue our study ofgauge-equivariant K-theory. In particular, we introduce and study products,which helps us establish the Thom isomorphism in gauge equivariant K-theory.Then we construct push-forward maps and define the topological index of agauge-invariant family.

Contents

1. Introduction 12. Preliminaries 32.1. Bundles of compact groups and finite holonomy conditions 32.2. Gauge-equivariant K-theory 42.3. Additional results 63. K-theory and complexes 83.1. The LnG–groups 83.2. Euler characteristics 103.3. Globally defined complexes 143.4. The non-compact case 154. The Thom isomorphism 154.1. Products 164.2. The non-compact case 175. Gysin maps 206. The topological index 24References 29

1. Introduction

In this paper we establish a Thom isomorphism theorem for gauge equivariantK-theory. Let p : G → B be a bundle of compact groups. Recall that this meansthat each fiber Gb := p−1(b) is a compact group and that, locally, G is of the form

V. N. was partially supported by NSF Grants DMS-9971951, DMS 02-00808, and a “collabo-rative research” grant. E. T. was partially supported by RFFI Grant 05-01-00923, Grant for thesupport of leading scientific schools and Grant “Universities of Russia” YP.04.02.530. The presentjoint research was started under the hospitality of MPIM (Bonn).

1

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2 VICTOR NISTOR AND EVGENIJ TROITSKY

U ×G, where U ⊂ B open and G a fixed compact group. Let X and B be locallycompact spaces and πX : X → B be a continuous map. In the present paper, asin [15], this map will be supposed to be a locally trivial bundle. A part of presentresults can be extended to the case of a general map, this, as well as the proof of ageneral index theorem, will be the subject of a forthcoming paper.

Assume that G acts on X . This action will be always be fiber-preserving. Thenwe can associate to the action of G on X G-equivariant K-theory groups K i

G(X) asin [15]. We shall review and slightly generalize this definition in Section 2.

For X compact, the group K0G(X) is defined as the Grothendieck group of

G-equivariant vector bundles on X . If X is not compact, we define the groupsK0

G(X) using fiberwise one-point compactifications. We shall call these groups sim-ply gauge-equivariant K-theory groups of X when we do not want to specify G. Thereason for introducing the gauge-equivariant K-theory groups is that they are thenatural range for the index of a gauge-invariant families of elliptic operators. Inturn, the motivation for studying gauge-invariant families and their index is due totheir connection to spectral theory and boundary value problems on non-compactmanifolds. Some possible connections with Ramond-Ramond fields in String The-ory were mentioned in [8, 15]. See also [1, 9, 12, 13].

In this paper, we continue our study of gauge-equivariant K-theory. We beginby providing two alternative definitions of the relative KG–groups, both based oncomplexes of vector bundles. (In this paper, all vector bundles are complex vectorbundles, with the exception of the tangent bundles and where explicitly stated.)These alternative definitions, modeled on the classical case [2, 10], provide a conve-nient framework for the study of products, especially in the relative or non-compactcases. The products are especially useful for the proof of the Thom isomorphismin gauge-equivariant theory, which is one of the main results of this paper. LetE → X be a G-equivariant complex vector bundle. Then the Thom isomorphismis a natural isomorphism

(1) τE : KiG(X) → Ki

G(E).

(There is also a variant of this result for spinc-vector bundles, but since we willnot need it for the index theorem [14], we will not discuss this in this paper.) TheThom isomorphism allows us to define Gysin (or push-forward) maps in K-theory.As it is well known from the classical work of Atiyah and Singer [4], the Thomisomorphism and the Gysin maps are some of the main ingredients used for thedefinition and study of the topological index. In fact, we shall proceed along thelines of that paper to define the topological index for gauge-invariant families ofelliptic operators. Some other approaches to Thom isomorphism in general settingsof Noncommutative geometry were the subject of [6, 7, 11, 16, 12] and many otherpapers.

Gauge-equivariant K-theory behaves in many ways like the usual equivariantK-theory, but exhibits also some new phenomena. For example, the groups K0

G(B)

may turn out to be reduced toK0(B) when G has “a lot of twisting” [15, Proposition3.6]. This is never the case in equivariant K-theory when the action of the group istrivial but the group itself is not trivial. In [15], we addressed this problem in twoways: first, we found conditions on the bundle of groups p : G → B that guaranteethat K0

G(X) is not too small (this condition is called finite holonomy and is recalled

below), and, second, we studied a substitute ofK0G(X) which is never too small (this

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THOM ISOMORPHISM 3

substitute is K(C∗(G)), the K-theory of the C∗-algebra of the bundle of compactgroups G).

In this paper, we shall again need the finite holonomy condition, so let us reviewit now. To define the finite holonomy condition, we introduced the representation

covering of G, denoted G → B. As a space, G is the union of all the representation

spaces Gb of the fibers Gb of the bundle of compact groups G. One measure of the

twisting of the bundle G is the holonomy associated to the covering G → B. We say

that G has representation theoretic finite holonomy if G is a union of compact-opensubsets. (An equivalent conditions can be obtained in terms of the fundamentalgroups when B is path-connected, see Proposition 2.3 below.)

Let C∗(G) be the enveloping C∗-algebra of the bundle of compact groups G. Weproved in [15, Theorem 5.2] that

(2) KjG(B) ∼= Kj(C

∗(G)),

provided that G has representation theoretic finite holonomy. This guarantees thatKj

G(B) is not too small. It also points out an alternative, algebraic definition of the

groups KiG(X).

The structure of the paper is as follows. We start from the definition of gauge-equivariant K-theory and some basic results from [15], most of them related tothe “finite holonomy condition,” a condition on bundles of compact groups thatwe recall in Section 2. In Section 3 we describe an equivalent definition of gauge-equivariant K-theory in terms of complexes of vector bundles. This will turn out tobe especially useful when studying the topological index. In Section 4 we establishthe Thom isomorphism in gauge-equivariant K-theory, and, in Section 5, we defineand study the Gysin maps. The properties of the Gysin maps allow us to define inSection 6 the topological index and establish its main properties.

2. Preliminaries

We now recall the definition of gauge-equivariantK-theory and some basic resultsfrom [15]. An important part of our discussion is occupied by the discussion of thefinite holonomy condition for a bundle of compact groups p : G → B.

All vector bundles in this paper are assumed to be complex vector bundles, unlessotherwise mentioned and excluding the tangent bundles to the various manifoldsappearing below.

2.1. Bundles of compact groups and finite holonomy conditions. We beginby introducing bundles of compact and locally compact groups. Then we studyfinite holonomy conditions for bundles of compact groups.

Definition 2.1. Let B be a locally compact space and let G be a locally compactgroup. We shall denote by Aut(G) the group of automorphisms of G. A bundle oflocally compact groups G with typical fiber G over B is, by definition, a fiber bundleG → B with typical fiber G and structural group Aut(G).

We fix the above notation. Namely, from now on and throughout this paper,unless explicitly otherwise mentioned, B will be a compact space and G → B willbe a bundle of compact groups with typical fiber G.

We need now to introduce the representation theoretic holonomy of a bundle ofLie group with compact fibers p : G → B. Let Aut(G) be the group of automor-phisms of G. By definition, there exists then a principal Aut(G)-bundle P → B

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such that

G ∼= P ×Aut(G) G := (P ×G)/Aut(G).

We shall fix P in what follows.Let G be the (disjoint) union of the sets Gb of equivalence classes of irreducible

representations of the groups Gb. Using the natural action of Aut(G) on G, we can

naturally identify G with P ×Aut(G) G as fiber bundles over B.Let Aut0(G) be the connected component of the identity in Aut(G). The group

Aut0(G) will act trivially on the set G, because the later is discrete. Let

HR := Aut(G)/Aut0(G), P0 := P/Aut0(G), and G ' P0 ×HRG.

Above, G is defined because P0 is an HR-principal bundle. The space G will be

called the representation space of G and the covering G → B will be called therepresentation covering associated to G.

Assume now that B is a path-connected, locally simply-connected space and fixa point b0 ∈ B. We shall denote, as usual, by π1(B, b0) the fundamental group ofB. Then the bundle P0 is classified by a morphism

(3) ρ : π1(B, b0) → HR := Aut(G)/Aut0(G),

which will be called the holonomy of the representation covering of G.For our further reasoning, we shall sometimes need the following finite holonomy

condition.

Definition 2.2. We say that G has representation theoretic finite holonomy if

every σ ∈ G is contained in a compact-open subset of G.

In the cases we are interested in, the above condition can be reformulated asfollows [15]

Proposition 2.3. Assume that B is path-connected and locally simply-connected.

Then G has representation theoretic finite holonomy if, and only if π1(B, b0)σ ⊂ Gis a finite set for any irreducible representation σ of G.

From now on we shall assume that G has representation theoretic finite holonomy.

2.2. Gauge-equivariant K-theory. Let us now define the gauge equivariant K-theory groups of a “G-fiber bundle” πY : Y → B. All our definitions are well knownif B is reduced to a point (cf. [2, 10]). First we need to fix the notation.

If fi : Yi → B, i = 1, 2, are two maps, we shall denote by

(4) Y1 ×B Y2 := {(y1, y2) ∈ Y1 × Y2, f1(y1) = f2(y2) }

their fibered product. Let p : G → B be a bundle of locally compact groups and letπY : Y → B be a continuous map. We shall say that G acts on Y if each group Gbacts continuously on Yb := π−1(b) and the induced map µ

G ×B Y := {(g, y) ∈ G × Y, p(g) = πY (y)} 3 (g, y) −→ µ(g, y) := gy ∈ Y

is continuous. If G acts on Y , we shall say that Y is a G-space. If, in additionto that, Y → B is also locally trivial, we shall say that Y is a G-fiber bundle, or,simply, a G-bundle. This definition is a particular case of the definition of the actionof a differentiable groupoid on a space.

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THOM ISOMORPHISM 5

Let πY : Y → B be a G-space, with G a bundle of compact groups over B.Recall that a vector bundle πE : E → Y is a G-equivariant vector bundle (or simplya G–equivariant vector bundle) if

πE := πY ◦ πE : E → B

is a G-space, the projection

πE : Eb := π−1E (b) → Yb := π−1

Y (b)

is Gb := p−1(b) equivariant, and the induced action Ey → Egy of g ∈ G, betweenthe corresponding fibers of E → Y , is linear for any y ∈ Yb, g ∈ Gb, and b ∈ B.

To define gauge-equivariantK–theory, we first recall some preliminary definitionsfrom [15]. Let πE : E → Y be a G-equivariant vector bundle and let πE′ : E′ → Y ′

be a G′-equivariant vector bundle, for two bundles of compact groups G → B andG′ → B′. We shall say that (γ, ϕ, η, ψ) : (G ′, E′, Y ′, B′) → (G, E, Y,B) is a γ–equivariant morphism of vector bundles if the following five conditions are satisfied:

(i) γ : G′ → G, ϕ : E′ → E, η : Y ′ → Y, and ψ : B → B′,(ii) all the resulting diagrams are commutative,(iii) ϕ(ge) = γ(g)ϕ(e) for all e ∈ E ′

b and all g ∈ G′b,,

(iv) γ is a group morphism in each fiber, and(v) f is a vector bundle morphism.

We shall say that φ : E → E ′ is a γ–equivariant morphism of vector bundles if,by definition, it is part of a morphism (γ, ϕ, η, ψ) : (G ′, E′, Y ′, B′) → (G, E, Y,B).Note that η and ψ are determined by γ and φ.

As usual, if ψ : B′ → B is a continuous [respectively, smooth] map, we define theinverse image (ψ∗(G), ψ∗(E), ψ∗(Y ), B′) of a G-equivariant vector bundle E → Yby ψ∗(G) = G ×B B′, ψ∗(E) = E ×B B′, and ψ∗(Y ) = Y ×B B′. If B′ ⊂ B and ψis the embedding, this construction gives the restriction of a G-equivariant vectorbundle E → Y to a closed, invariant subset B′ ⊂ B of the base of G, yielding aGB′–equivariant vector bundle. Usually G will be fixed, however.

Let p : G → B be a bundle of compact groups and πY : Y → B be a G–space.The set of isomorphism classes of G–equivariant vector bundles πE : E → Y willbe denoted by EG(Y ). On this set we introduce a monoid operation, denoted “+,”using the direct sum of vector bundles. This defines a monoid structure on the setEG(Y ) as in the case when B consists of a point.

Definition 2.4. Let G → B be a bundle of compact groups acting on the G-spaceY → B. Assume Y to be compact. The G-equivariant K-theory group K0

G(Y ) isdefined as the group completion of the monoid EG(Y ).

When working with gauge-equivariant K–theory, we shall use the following ter-minology and notation. If E → Y is a G-equivariant vector bundle on Y , we shalldenote by [E] its class in K0

G(Y ). Thus K0G(Y ) consists of differences [E] − [E1].

The groups K0G(Y ) will also be called gauge equivariant K-theory groups, when we

do not need to specify G. If B is reduced to a point, then G is group, and the groupsK0

G(Y ) reduce to the usual equivariant K-groups.We have the following simple observations on gauge-equivariantK–theory. First,

the familiar functoriality properties of the usual equivariantK-theory groups extendto the gauge equivariant K-theory groups. For example, assume that the bundle ofcompact groups G → B acts on a fiber bundle Y → B and that, similarly, G ′ → B′

acts on a fiber bundle Y ′ → B′. Let γ : G → G′ be a morphism of bundles of

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6 VICTOR NISTOR AND EVGENIJ TROITSKY

compact groups and f : Y → Y ′ be a γ-equivariant map. Then we obtain a naturalgroup morphism

(5) (γ, f)∗ : K0G′(Y ′) → K0

G(Y ).

If γ is the identity morphism, we shall denote (γ, f)∗ = f∗.A G-equivariant vector bundle E → Y on a G-space Y → B, Y compact, is called

trivial if, by definition, there exists a G-equivariant vector bundle E ′ → B such thatE is isomorphic to the pull-back of E ′ to Y . Thus E ' Y ×B E

′. If G → B hasrepresentation theoretic finite holonomy and Y is a compact G-bundle, then everyG–equivariant vector bundle over Y can be embedded into a trivial G–equivariantvector bundle. This embedding will necessarily be as a direct summand.

If G → B does not have finite holonomy, it is possible to provide examples ifG–equivariant vector bundles that do not embed into trivial G–equivariant vectorbundles [15]. Also, a related example from [15] shows that the groups K0

G(Y ) canbe fairly small if the holonomy of G is “large.”

A further observation is that it follows from the definition that the tensor productof vector bundles defines a natural ring structure on K0

G(Y ). We shall denote theproduct of two elements a and b in this ring by a⊗b or, simply, ab, when there is nodanger of confusion. In particular the groups K i

G(X) for πX : X → B are equipped

with a natural structure of K0G(B)–module obtained using the pull-back of vector

bundles on B, namely, ab := π∗X (a) ⊗ b ∈ K0

G(X) for a ∈ K0G(B) and b ∈ K0

G(X).The definition of the gauge-equivariant groups extends to non-compact G-spaces

Y as in the case of equivariant K–theory. Let Y be a G-bundle. We shall denotethen by Y + := Y ∪ B the compact space obtained from Y by the one-point com-pactification of each fiber (recall that B is compact). The need to consider thespace Y + is the main reason for considering also non longitudinally smooth fibersbundles on B. Then

K0G(Y ) := ker

(K0

G(Y +) → K0G(B)

).

Also as in the classical case, we let

KnG (Y, Y ′) := K0

G((Y \ Y ′) × Rn)

for a G-subbundle Y ′ ⊂ Y . Then [15] we have the following periodicity result

Theorem 2.5. We have natural isomorphisms

KnG (Y, Y ′) ∼= Kn−2

G (Y, Y ′).

Gauge-equivariant K-theory is functorial with respect to open embeddings. In-deed, let U ⊂ X be an open, G-equivariant subbundle. Then the results of [15,Section 3] provide us with a natural map morphism

(6) i∗ : KnG (U) → Kn

G (X).

In fact, i∗ is nothing but the composition KnG (U) ∼= Kn

G (X,X \ U) → KnG (X).

2.3. Additional results. We now prove some more results on gauge-equivariantK-theory.

Let G → B and H → B be two bundles of compact groups over B. Recall thatan H-bundle πX : X → B is called free if the action of each group Hb on the fiberXb is free (i.e., hx = x, x ∈ Xb, implies that h is the identity of Gb.) We shallneed the following result, which is an extension of a result in [10, page 69]. Forsimplicity, we shall write G ×H instead of G ×B H.

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THOM ISOMORPHISM 7

Theorem 2.6. Suppose πX : X → B is a G × H-bundle that is free as an H-bundle. Let π : X → X/H be the (fiberwise) quotient map. For any G–equivariantvector bundle πE : E → X/H, we define the induced vector bundle

π∗(E) :={(x, ε) ∈ X ×E, π(x) = πE(ε)

}→ X,

with the action of G×H given by (g, h) · (x, ε) := ((g, h)x, gε). Then π∗ is gives riseto a natural isomorphism K0

G(X/H) → K0G×H(X).

Proof. Let π∗ : K0G(X/H) → K0

G×H(X) be the induction map, as above. We will

construct a map r : K0G×H(X) → K0

G(X/H) satisfying π∗ ◦ r = Id and r ◦ π∗ = Id.Let πF : F → X be a G ×H-vector bundle. Since the action of H on X is free, theinduced map πF : F/H → X/H of quotient spaces is a (locally trivial) G-bundle.Clearly, this construction is invariant under homotopy, and hence we can definer[F ] := [F/H].

Let us check now that r is indeed an inverse of π∗. Denote by F 3 f → Hf ∈F/H the quotient map. Let F → X be a G ×H-vector bundle. To begin with, thetotal space of π∗ ◦ r(F ) is

{(x,Hf) ∈ X × (F/H), π∗(x) = πF (Hf)

},

by definition. Then the map F 3 f → (πF (f),Hf) ∈ π∗ ◦ r(F ) is an isomorphism.Hence, π∗r = Id.

Next, consider a G–vector bundle πE : E → X/H. The total space of r ◦ π∗(E)is then {

(Hy, ε) ∈ (X/H)×E, Hy = πE(ε))},

because H acts only on the first component of π∗(E). Then

E → r ◦ π∗(E), ε 7→ (π(ε), ε)

is an isomorphism. Hence, r ◦ π∗ = Id. �

See also [15, Theorem 3.5].

Corollary 2.7. Let P be a G ×H–bundle that is free as an H–bundle. Also, letW be an H–bundle, then there is a natural isomorphism

KG×H(P ×B W ) ∼= KG(P ×H W ).

Proof. Take X := P ×B W in the previous theorem. Then X is a free H–bundle,because P is, and X/H =: P ×H W . �

In the following section, we shall also need the following quotient constructionassociated to a trivialization of a vector bundle over a subset. Namely, if Y ⊂ Xis a G–invariant, closed subbundle, then we shall denote by X/BY the fiberwisequotient space over B, that is the quotient of X with respect to the equivalencerelation ∼, x ∼ y if, and only if, x, y ∈ Yb, for some b ∈ B.

If E is a G–equivariant vector bundle over a G–bundle X , together with a G–equivariant trivialization over a G–subbundle Y ⊂ X , then we can generalize thequotient (or collapsing) construction of [2, §1.4] to obtain a vector bundle overX/BY , where by X/BY we denote the fiberwise quotient bundle over B, as above.

Lemma 2.8. Suppose that X is a G–bundle and that Y ⊂ X is a closed, G–invariant subbundle. Let E → X be a G–vector bundle and α : E|Y ∼= Y ×B V be aG–equivariant trivialization, where V → B is a G–equivariant vector bundle. Then

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8 VICTOR NISTOR AND EVGENIJ TROITSKY

we can naturally associate to (E,α) a naturally defined vector bundle E/α→ X/BYthat depends only on homotopy class of α.

Proof. Let p : Y ×B V → V be the natural projection. Introduce the followingequivalence relation on E:

e ∼ e′ ⇔ e, e′ ∈ E|Y and pα(e) = pα(e′).

Let then E/α be equal to E/ ∼. This is locally trivial vector bundle over X/BY .Indeed, it is necessary to verify this only in a neighborhood of Y/BY ∼= B. Let Ube a G invariant open subset of X such that α can be extended to an isomorphismα : E|U ∼= U ×B V . We obtain an isomorphism

α′ : (E|U )/α ∼= (U/BY ) ×B V , α′(e) = α(e).

Moreover, (U/BY ) ×B V is a locally trivial G–equivariant vector bundle.Suppose that α0 and α1 are homotopic trivializations of E|Y , that is, trivi-

alizations such that there exists a trivialization β : E × I |Y×I∼= Y × I ×B V ,

β|E×{0} = α0 and β|E×{1} = α1. Let

f : X/BY × I → (X × I)/(B×I)(Y × I).

Then the bundle f∗((E×I)/β) overX/BY ×I satisfies f∗((E×I)/β)|(X/BY )×{i} =E/αi, i = 0, 1. Hence, E/α0

∼= E/α1. �

3. K-theory and complexes

For the purpose of defining the Thom isomorphism, it is convenient to workwith an equivalent definition of gauge-equivariant K-theory in terms of complexesof vector bundles. This will turn out to be especially useful when studying thetopological index.

The statements and proofs of this section, except maybe Lemma 3.4, follow theclassical ones [2, 10], so our presentation will be brief.

3.1. The LnG–groups. We begin by adapting some well known concepts and con-structions to our settings.

Let X → B be a locally compact, paracompact G–bundle. A finite complex ofG–equivariant vector bundles over X is a complex

(E∗, d) =(. . .

di−1−→ Ei

di−→ Ei+1 di+1−→ . . .

)

i ∈ Z, of G–equivariant vector bundles over X with only finitely many E i’s differentfrom zero. Explicitly, Ei are G–equivariant vector bundles, di’s are G–equivariantmorphisms, di+1di = 0 for every i, and Ei = 0 for |i| large enough. We shall alsouse the notation (E∗, d) =

(E0, . . . , En, di : Ei|Y → Ei+1|Y

),, if Ei = 0 for i < 0

and for i > n.As usual, a morphism of complexes f : (E∗, d) → (F ∗, δ) is a sequence of mor-

phisms fi : Ei → F i such that fi+1di = δi+1fi, for all i. These constructions yieldthe category of finite complexes of G–equivariant vector bundles. Isomorphism inthis category will be denoted by (E∗, d) ∼= (F ∗, δ).

In what follows, we shall consider a pair (X,Y ) of G–bundles withX is a compactG–bundle, unless explicitly otherwise mentioned.

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THOM ISOMORPHISM 9

Definition 3.1. Let X be a compact G–bundle and Y be a closed G–invariantsubbundle. Denote by CnG (X,Y ) the set of (isomorphism classes of) sequences

(E∗, d) =(E0, E1, . . . , En, dk : Ek|Y → Ek+1|Y

)

of G–equivariant vector bundles over X such that (Ek|Y , d) is exact if we let Ej = 0for j < 0 or j > n.

We endow CnG (X,Y ) with the semigroup structure given by the direct sums ofcomplexes. An element in CnG (X,Y ) is called elementary if it is isomorphic to acomplex of the form

. . .→ 0 → EId−→ E → 0 → . . . ,

Two complexes (E∗, d), (F ∗, δ) ∈ CnG (X,Y ) are called equivalent if, and only if,

there exist elementary complexes Q1, . . . , Qk, P 1, . . . , Pm ∈ CnG (X,Y ) such that

E ⊕Q1 ⊕ · · · ⊕Qk ∼= F ⊕ P1 ⊕ · · · ⊕ Pm.

We write E ' F in this case. The semigroup of equivalence classes of sequences inCnG (X,Y ) will be denoted by LnG(X,Y ).

We obtain, from definition, natural injective semigroup homomorphisms

CnG (X,Y ) → Cn+1G (X,Y ) and CG(X,Y ) :=

n

CnG (X,Y ).

The equivalence relation ∼ commutes with embeddings, so the above morphismsinduce morphisms LnG(X,Y ) → Ln+1

G (X,Y ). Let L∞G (X,Y ) := lim

→LnG(X,Y ).

Lemma 3.2. Let E → X and F → X be G–vector bundles. Let α : E|Y → F |Yand β : E → F be surjective morphisms of G–equivariant vector bundles. Also,assume that α and β|Y are homotopic in the set of surjective G–equivariant vectorbundle morphisms. Then there exists a surjective morphism of G–equivariant vectorbundles α : E → F such that α|Y = α. The same result remains true if we replace“surjective” with “injective” or “isomorphism” everywhere.

Proof. Let Z := (Y × [0, 1]) ∪ (X × {0}) and π : Z → X be the projection. Letπ∗(E) → Z and π∗(F ) → Z be the pull-backs of E and F . The homotopy in thestatement of the Lemma defines a surjective morphism a : π∗(E) → π∗(F ) suchthat a|Y×{1} = α and a|X×{0} = β. By [15, Lemma 3.12], the morphism a can beextended to a surjective morphism over (U × [0, 1])∪ (X×{0}), where U is an openG–neighborhood of Y . (In fact, in that Lemma we considered only the case of anisomorphism, but the case of a surjective morphism is proved in the same way.) Letϕ : X → [0, 1] be a continuous function such that ϕ(Y ) = 1 and ϕ(X \ U) = 0. Byaveraging, we can assume ϕ to be G–equivariant. Then define α(x) = a(x, ϕ(x)),for all x ∈ X . �

Remark 3.3. Suppose that X is a compact G–space and Y = ∅. Then we havea natural isomorphism χ1 : L1

G(X, ∅) → K0G(X) taking the class of E1, E0 to the

element [E0] − [E1].

We shall need the following lemma.

Lemma 3.4. Let p : G → B be a bundle of compact groups and πX : X → B bea compact G–bundle. Assume that πX has a cross-section, which we shall use toidentify B with a subset of X. Then the sequence

0 → L1G(X,B) → L1

G(X) → L1G(B)

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10 VICTOR NISTOR AND EVGENIJ TROITSKY

is exact.

Proof. Suppose that E = (E1, E0, ϕ) defines an element of L1G(X) such that its

image in L1G(B) is zero. Then the definition of E ∼ 0 in L1

G(B) shows that the

restrictions of E1 and E0 to B are isomorphic over B. Hence, the above sequenceis exact at L1

G(X).

Suppose now that (E1, E0, ϕ) represents a class in L1G(X,B) such that its image

in L1G(X) is zero. This means (keeping in mind Remark 3.3) that there exists a G–

equivariant vector bundle P and an isomorphism ψ : E1⊕P ∼= E0⊕P . Let us define

P := P ⊕ π∗X (E0|B) ⊕ π∗

X(P |B), where πX : X → B is the canonical projection, as

in the statement of the Lemma. Also, define ψ = ψ⊕ Id : E1 ⊕P → E0 ⊕P , whichis also an isomorphism.

We thus obtain that T := ψ(ϕ⊕ Id)−1 is an automorphism of (E0⊕P )|B . which

has the form

(β 00 Id

)with the respect to the decomposition

(E0 ⊕ P )|B = (E0 ⊕ P )|B ⊕ (E0 ⊕ P )|B .

The automorphism T := ψ(ϕ⊕ Id)−1 is homotopic to the automorphism T1 definedby the matrix (

Id 00 β

).

Since T1 extends to an automorphism of E0 ⊕ P over X , namely

(Id 00 π∗

X (β)

),

Lemma 3.2 gives that the automorphism ψ(ϕ ⊕ Id)−1 also can be extended to X ,such that over B we have the following commutative diagram:

(E1 ⊕ P )|Bϕ⊕Id //

ψ|B

��

(E0 ⊕ P )|B

α|B=

0@ β 0

0 Id

1A

��(E0 ⊕ P )|B

Id // (E0 ⊕ P )|B .

Hence, (E1, E0, ϕ)⊕(P, P, Id) ∼= (E0⊕P,E0⊕P, Id) and so is zero in L1G(X,B). �

3.2. Euler characteristics. We now generalize the above construction to othergroups LnG , thus proving the existence and uniqueness of Euler characteristics.

Definition 3.5. Let X be a compact G–space and Y ⊂ X be a G–invariant subset.An Euler characteristic χn is a natural transformation of functors χn : LnG(X,Y ) →K0

G(X,Y ), such that for Y = ∅ it takes the form

χn(E) =

n∑

i=0

(−1)i[Ei],

for any sequence E = (E∗, d) ∈ LnG(X,Y ).

Lemma 3.6. There exists a unique natural transformation of functors (i.e., anEuler characteristic)

χ1 : L1G(X,Y ) → K0

G(X,Y ),

which, for Y = ∅, has the form indicated in 3.3.

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THOM ISOMORPHISM 11

Proof. To prove the uniqueness, suppose that χ1 and χ′1 are two Euler charac-

teristics on L1G . Then χ′

1χ−11 is a natural transformation of K0

G that is equal to

the identity on each K0G(X). Let us consider the long exact sequence a long exact

sequence

(7) · · · → Kn−1G (Y, Y ′) → Kn−1

G (Y ) → Kn−1G1

(Y ′)

→ KnG (Y, Y ′) → Kn

G (Y ) → KnG (Y ′) → . . .

associated to a pair (Y, Y ) of G–bundles (see [15], Equation (10) for a proof ofthe exactness of this sequence). The map K0

G(X,B) → K0G(X) from this exact

sequence is induced by (X, ∅) → (X,B), and hence, in particular, it is natural.Assume that πX : X → B has a cross-section. Then the exact sequence (7) for(Y, Y ′) = (X,B) yields a natural exact sequence 0 → K0

G(X,B) → K0G(X). This in

conjunction with Lemma 3.4 shows that χ′1χ

−11 is the identity on K0

G(X,B). Recallnow that we agreed to denote by X/BY the fiberwise quotient space over B, that isthe quotient of X with respect to the equivalence relation ∼, x ∼ y if, and only if,x, y ∈ Yb, for some b ∈ B. Finally, since the map (X,Y ) → (X/BY,B) induces anisomorphism of K0

G-groups [15, Theorem 3.19], χ′1χ

−11 is the identity on K0

G(X,Y )for all pairs (X,Y ).

To prove the existence of the Euler characteristic χ1, let (E1, E0, α) := (α :E1 → E0) represent an element of L1

G(X,Y ). Suppose that X0 and X1 are twocopies of X and Z := X0 ∪Y X1 → B is the G–bundle obtained by identifying thetwo copies of Y ⊂ Xi, i = 0, 1. The identification of E1|Y and E0|Y with the helpof α gives rise to an element [F 0] − [F 1] ∈ K0

G(Z) defined as follows. By adding

some bundle to both Ei’s, we can assume that E1 is trivial (that is, it is isomorphicto the pull–back of a vector bundle on B). Then E1 extends to a trivial G–vector

bundle E1 → Z. We define F 0 := E0 ∪α E1 and F 1 := E1 .The exact sequence (7) and the natural G-retractions πi : Z → Xi, give natural

direct sum decompositions

(8) K0G(Z) = K0

G(Z,Xi) ⊕K0G(Xi), i = 0, 1.

The natural map (X0, Y ) → (Z,X1) induces an isomorphism

k : K0G(Z,X1) → K0

G(X0, Y ).

Let us define then χ1(E0, E1, α) to be equal to the image under k of the K0

G(Z,X1)-

component of [E1, α, E0] (with the respect to (8)). It follows from its definitionthat this map is natural, respects direct sums, and is independent with respect tothe addition of elementary elements. Our proof is completed by observing thatχ1(E

1, E0, α) = [E0] − [E1] when Y = ∅. �

We shall also need the following continuity property of the functor L1G . Recall

that we have agreed to denote by X/BY the fiberwise quotient bundle over B.

Lemma 3.7. The natural homomorphism

Π∗ : L1G(X/BY, Y/BY ) = L1

G(X/BY,B) → L1G(X,Y )

is an isomorphism for all pairs (X,Y ) of compact G-bundles.

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12 VICTOR NISTOR AND EVGENIJ TROITSKY

Proof. Lemmata 3.6 and 3.4 give the following commutative diagram

L1G(X/BY,B)

Π∗

//

∼= χ1

��

L1G(X,Y )

χ1

��K0

G(X/BY,B)Π∗

∼=// K0

G(X,Y ).

From this we obtain injectivity.To prove surjectivity, suppose that E1 and E0 are G–equivariant vector bundles

over X and α : E1|Y → E0|Y is an isomorphism of the restrictions. Let P → X bea G-bundle such that there is an isomorphism β : E1 ⊕ P ∼= F , where F is a trivialbundle (i.e., isomorphic to a pull back from B). Then (E1, E0, α) ∼ (F,E0 ⊕P, γ),where γ = (α ⊕ Id)β−1. The last object is the image of (F, (E0 ⊕ P )/γ, γ/γ) (seeLemma 2.8). �

We obtain the following corollaries.

Corollary 3.8. The Euler characteristic χ1 : L1G(X,Y ) → K0

G(X,Y ) is an iso-morphism and hence it is defines an equivalence of functors.

Proof. This follows from Lemmas 3.7 and 3.6. �

Lemma 3.9. The class of (E1, E0, α) in L1G(X,Y ) depends only on the homotopy

class of the isomorphism α.

Proof. Let Z = X × [0, 1], W = Y × [0, 1]. Denote by p : Z → X the natural pro-jection and assume that αt is a homotopy, where α0 = α. Then αt gives rise an iso-morphism β : p∗(E1)|W ∼= p∗(E0)|W , and hence to an element (p∗(E1), p∗(E0), β)of L1

G(Z,W ). If

it : (X,Y ) → (X × {t}, Y × {t}) ⊂ (Z,W ),

are the standard inclusions, then (E1, E0, αt) = i∗t (p∗(E1), p∗(E0), β). Consider

the commutative diagram

L1G(X,Y )

χ1

��

L1G(Z,W )

χ1

��

i∗0oo i∗1 // L1G(X,Y )

χ1

��K0

G(X,Y ) K0G(Z,W )

i∗0oo i∗1 // K0G(X,Y ).

The vertical morphisms and and the morphisms of the bottom line of the abovediagram are isomorphisms. Hence, the arrows of the top line are isomorphisms too.The composition i∗0(i

∗1)

−1 is identity for the bottom line, hence it is the identity forthe top line too. �

The following theorem reduces the study of the functors LnG , n > 1, to the study

of L1G .

Theorem 3.10. The natural map jn : LnG(X,Y ) → Ln+1G (X,Y ) is an isomor-

phism.

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THOM ISOMORPHISM 13

Proof. Let E = (E0, E1, . . . , En+1; dk), dk : Ek|Y → Ek+1|Y represent an elementof the semigroup Ln+1

G (X,Y ). To prove the surjectivity of jn, let us first noticethat E is equivalent to the complex

(E0, . . . , En−2, En−1 ⊕En+1, En ⊕En+1, En+1;

d0, . . . , dn−2 ⊕ 0, dn−1 ⊕ Id, dn ⊕ 0).

The maps dn ⊕ 0 : (En⊕En+1)|Y → En+1|Y and 0⊕ Id : (En⊕En+1)|Y → En+1|Yare homotopic within the set of surjective, G-equivariant vector bundle morphisms(En ⊕ En+1)|Y → En+1|Y . Hence, by Lemma 3.2, dn ⊕ 0 can be extended to asurjective morphism b : En ⊕ En+1 → En+1 of G–equivariant vector bundles (overthe whole of X). So, the bundle En⊕En+1 is isomorphic to ker(b)⊕En+1. Hence,the E is equivalent to

(E0, . . . , En−2, En−1 ⊕En+1, ker(b), 0; d0, . . . , dn−2 ⊕ 0, dn−1, 0).

This proves the surjectivity of jn.To prove the injectivity of jn, it is enough to define, for any n, a left inverse

qn : LnG(X,Y ) → L1G(X,Y ) to sn := jn−1 ◦ · · · ◦ j1. Suppose that (E∗; d) represents

an element of semigroup LnG(X,Y ). Choose G-invariant Hermitian metrics on Ei

and let d∗i : Ei+1|Y → Ei|Y be the adjoint of di. Let

F 0 :=⊕

i

E2i, F 1 :=⊕

i

E2i+1, b : F 0|Y → F 1|Y , b =∑

i

(d2i + d∗2i+1).

A standard verification shows that b is an isomorphism. Since all invariant metricsare homotopic to each other, Lemma 3.9 shows that (E, d) → (F, b) defines amorphism qn : LnG(X,Y ) → L1

G(X,Y ). This is the desired left inverse for sn. �

Let us observe that the proof of the above theorem and Lemma 3.9 give thefollowing corollary.

Corollary 3.11. The class of E = (Ei, di) in LnG(X,Y ) does not change if wedeform the differentials di continuously.

We are now ready to prove the following basic result.

Theorem 3.12. For each n there exists a unique Euler characteristic

χn : LnG(X,Y ) ∼= K0G(X,Y ).

In particular, L∞G (X,Y ) ∼= K0

G(X,Y ) and LnG(X,Y ) has a natural group structurefor any closed, G–invariant subbundle Y ⊂ X.

Proof. The statement is obtained from the lemmas we have proved above as follows.First of all, Theorem 3.10 allows us to define

χn := χ1 ◦ j−11 ◦ . . . j−1

n−1 : LnG(X,Y ) → K0G(X,Y ).

Lemma 3.7 shows that χn is an isomorphism. The uniqueness of χn is proved inthe same way as the uniqueness of χ1 (Lemma 3.6). �

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14 VICTOR NISTOR AND EVGENIJ TROITSKY

3.3. Globally defined complexes. The above theorem provides us with an al-ternative definition of the groups K0

G(X,Y ). We now derive yet another definitionof these groups that is closer to what is needed in applications and is based ondifferentials defined on X , not just on Y .

Let (E, d) be a complex of G–equivariant vector bundles over a G–space X . Apoint x ∈ X will be called a point of acyclicity of (E, d) if the restriction of (E, d)to x, i.e., the sequence of linear spaces

(E, d)x =(. . .

(di)x−→ Eix

(di+1)x

−→ Ei+1x

(di+2)x

−→ . . .),

is exact. The support supp(E, d) of the finite complex (E, d) is the complement inX of the set of its points of acyclicity. This definition and the following lemma holdalso for X non-compact.

Lemma 3.13. The support supp(E, d) is a closed G–invariant subspace of X.

Proof. The fact that supp(E, d) is closed is classical (see [2, 10] for example). Theinvariance should be checked up over one fiber of X at b ∈ B. But this is onceagain a well known fact of equivariant K-theory (see e.g. [10]). �

Lemma 3.14. Let En, . . . , E0 be G–equivariant vector bundles over X and Ybe a closed, G–invariant subbundle of X. Suppose there are given morphisms di :Ei|Y → Ei−1|Y such that (Ei|Y , di) is an exact complex. Then the morphisms dican be extended to morphisms defined over X such that we still have a complex ofG–equivariant vector bundles.

Proof. We will show that we can extend each di to a morphism ri : Ei → Ei−1

such that ri−1 ◦ ri = 0. Let us find a G-invariant open neighborhood U of Y in Xsuch that for any i there exist an extension si of di to U with (E, s) still an exactsequence. The desired ri will be then be defined as ri = ρsi, where ρ : X → [0, 1]is a continuous function, ρ = 1 on Y and supp ρ ⊂ U .

Let us construct U by induction over i. Assume that for the closure U i of someopen G-neighborhood of Y in X we can extend dj to sj (j = 1, . . . , i) such that on

U i the sequence

Eisi−→ Ei−1 si−1

−→ · · · → E0 → 0

is exact. Suppose, Ki := ker(si|U i). Then di+1 determines a cross section of thebundle HomG(Ei,Ki)|Y . This section can be extended to an open G-neighborhoodV of Y in U i. We hence obtain an extension si+1 : Ei+1 → Ki of di+1 : Ei+1 → Ki

over V . Since di+1|Y is surjective (with range Ki), the morphism si+1 will besurjective U i+1 for some open Ui+1 ⊂ Ui. �

The above lemma suggests the following definition.

Definition 3.15. Let X be a compact G–bundle and Y ⊂ X be a G–invariant sub-bundle. We define EnG (X,Y ) to be the semigroup of homotopy classes of complexesof G–equivariant vector bundles of length n over X such that their restrictions toY are acyclic (i.e., exact).

We shall say that two complexes are homotopic if they are isomorphic to therestrictions to X × {0} and X × {1} of a complex defined over X × I and acyclicover Y × I .

Remark 3.16. By Corollary 3.11, the restriction of morphisms induces a morphismΦn : EnG (X,Y ) → LnG(X,Y ).

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THOM ISOMORPHISM 15

Theorem 3.17. Let X be a compact G–bundle and Y ⊂ X be a G–invariantsubbundle. Then the natural transformation Φn, defined in the above remark, is anisomorphism.

Proof. The surjectivity of Φn follows from 3.14. The injectivity of Φn can be provedin the same way as [2, Lemma 2.6.13], keeping in mind Lemma 3.14.

More precisely, we need to demonstrate that differentials of any complex overG-subbundle (X ×{0})∪ (X ×{1})∪ (Y × I) of X × I , which is acyclic over Y × I ,can be extended to a complex over the entire X × I . The desired construction hasthe following three stages. First, let V be a G-invariant neighborhood of Y suchthat the restriction of our complex is still acyclic on (V ×{0})∪ (V ×{1})∪ (Y × I)as well as on its closure (V × {0}) ∪ (V × {1}) ∪ (Y × I). By Lemma 3.14, onecan extend the differentials di to G–equivariant morphisms ri over V × I that stilldefine a complex. Second, let ρ1, ρ2 be a G-invariant partition of unity subordinatedto covering V × I , (X \ Y ) × I of X × Y . Let us extend original differentials to(X× [0, 1/4])∪ (X× [3/4, 1])∪ (V ×I) by taking di(x, t) := di(x, 0) for t ≤ 1

4 ·ρ2(x),x ∈ X \ Y . Similarly near t = 1. Also,

di(x, t) := ri

(x,

(t− 1

4 · ρ2(x))

(1 − 1

2 · ρ2(x))), x ∈ V.

Third, multiplying this differential by a function τ : X × I → I equal to 1 on theoriginal subset of definition of the differential and to 0 off (X × [0, 1/4]) ∪ (X ×[3/4, 1])∪ (V × I) we obtain the desired extension. �

3.4. The non-compact case. In the case of a locally compact, paracompact G–bundle X , we change the definitions of LnG and EnG as follows. In the definitionof LnG , the morphisms di have to be defined and form an exact sequence off theinterior of some compact G–invariant subset C of X r Y (the complement of Y inX). In the definition of EnG , the complexes have to be exact outside some compactG–invariant subset of X r Y . In other words, LnG(X,Y ) = LnG(X+, Y +).

Since the proof of Lemma 3.14 is still valid, we have the analogue of Theo-rem 3.17: there is a natural isomorphism

LnG(X,Y ) ∼= EnG (X,Y ).

The proof of the other statement also can be extended to the non-compact case.The only difference is that we have to replace Y with X \U , where U is an open, G–invariant subset with compact closure. Then, when we study two element sequencesE = (Ei, di), we have to take the unions of the corresponding open sets. Of course,these sets are not bundles, unlike Y , but for our argument using extensions this isnot dangerous. This ultimately gives

(9) K0G(X,Y ) ∼= LnG(X,Y ) ∼= EnG (X,Y ), n ≥ 1.

As we shall see below, the liberty of using these equivalent definitions of K0G(X,Y )

is quite convenient in applications, especially when studying products.

4. The Thom isomorphism

In this section, we establish the Thom isomorphism in gauge-equivariant K-theory. We begin with a discussion of products and of the Thom morphism.

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16 VICTOR NISTOR AND EVGENIJ TROITSKY

4.1. Products. Let πX : X → B be a G–space, πF : F → X be a complex G–vector bundle over X , and s : X → F a G–invariant section. We shall denote byΛiF the i-th exterior power of F , which is again a complex G–equivariant vectorbundle over X . As in the proof of the Thom isomorphism for ordinary vectorbundles, we define the complex Λ(F, s) of G–equivariant vector bundles over X by

(10) Λ(F, s) := (0 → Λ0Fα0

−→ Λ1Fα1

−→ . . .αn−1

−→ ΛnF → 0),

where αk(vx) = s(x) ∧ vx for vx ∈ ΛkF x and n = dimF . It is immediate to checkthat αj+1(x)αj(x) = 0, and hence that (Λ(F, s), α) is indeed a complex.

The Kunneth formula shows that the complex Λ(F, s) is acyclic for s(x) 6= 0, andhence supp(Λ(F, s)) := {x ∈ X |s(x) = 0}. If this set is compact, then the resultsof Section 3 will associate to the complex Λ(F, s) of Equation (10) an element

(11) [Λ(F, s)] ∈ K0G(X).

Let X be a G–bundle and πF : F → X be a G–equivariant vector bundle overX . The point of the above construction is that π∗

F (F ), the lift of F back to itself,has a canonical section whose support is X . Let us recall how this is defined. LetπFF : π∗

F (F ) → F be the G–vector bundle over F with total space

π∗F (F ) := {(f1, f2) ∈ F × F, πF (f1) = πF (f2)}

and πFF (f1, f2) = f1. The vector bundle πFF : π∗F (F ) → F has the canonical

section

sF : F → π∗FF, sF (f) = (f, f).

The support of sF is equal to X . Hence, if X is a compact space, using again theresults of Section 3, especially 3.17, we obtain an element

(12) λF := [Λ(π∗F (F ), sF )] ∈ K0

G(F ).

Recall that the tensor product of vector bundles defines a natural product ab =a ⊗ b ∈ K0

G(X) for any a ∈ K0G(B) and any b ∈ K0

G(X), where πX : X → B is acompact G–space, as above.

Recall that all our vector bundles are assumed to be complex vector bundles,except for the ones coming from geometry (tangent bundles, their exterior powers)and where explicitly mentioned. Due to the importance that F be complex in thefollowing definition, we shall occasionally repeat this assumption.

Definition 4.1. Let πF : F → X be a (complex) G–equivariant vector bundle.Assume the G–bundle X → B is compact and let λF ∈ K0

G(F ) be the class definedin Equation (12), then the mapping

ϕF : K0G(X) → K0

G(F ), ϕF (a) = π∗F (a) ⊗ λF .

is called the Thom morphism.

As we shall see below, the definition of the Thom homomorphism extends to thecase when X is not compact, although the Thom element itself is not defined if Xis not compact.

The definition of the Thom isomorphism immediately gives the following propo-sition. We shall use the notation of Proposition 4.1.

Proposition 4.2. The Thom morphism ϕF : K0G(X) → K0

G(F ) is a morphism of

K0G(B)–modules.

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THOM ISOMORPHISM 17

Let ι : X ↪→ F be the zero section embedding of X into F . Then ι induceshomomorphisms

ι∗ : K0G(F ) → K0

G(X) and ι∗ ◦ ϕF : K0G(X) → K0

G(X).

It follows from the definition that ι∗ϕF (a) = a ·∑n

i=0(−1)iΛiF.

4.2. The non-compact case. We now consider now the case when X is locallycompact, but not necessarily compact. The complex Λ(π∗

F (F ), sF ) has a non-compact support, and hence it does not define an element of K0

G(F ). However, if

a = [(E,α)] ∈ K0G(X) is represented by the complex (E,α) of vector bundles with

compact support (Section 3), then we can still consider the tensor product complex(π∗F (E), π∗

F (α))⊗ Λ(π∗

FF, sF ).

From the Kunneth formula for the homology of a tensor product we obtain thatthe support of a tensor product complex is the intersection of the supports of thetwo complexes. In particular, we obtain

(13) supp{(π∗FE, π

∗Fα) ⊗ Λ(π∗

FF, sF )} ⊂ supp(π∗FE, π

∗Fα) ∩ supp Λ(π∗

FF, sF ) ⊂

⊂ supp(π∗FE, π

∗Fα) ∩X = supp(E,α).

Thus, the complex (π∗F E , π

∗Eα)⊗Λ(π∗

FF, sF ) has compact support and hence definesan element in K0

G(F ).

Proposition 4.3. The homomorphism of K0G(B)-modules

(14) ϕF : K0G(X) → K0

G(F ), ϕF (a) = [(π∗F E , π

∗Fα) ⊗ Λ(π∗

FF, sF )],

defined in Equation (13) extends the Thom morphism to the case of not necessarilycompact X. The Thom morphism ϕF satisfies

(15) i∗ϕF (a) = a ·n∑

i=0

(−1)iΛiF

in the non-compact case as well.

Let F → X be a G–equivariant vector bundle and F 1 = F × R, regarded asa vector bundle over X × R. The periodicity isomorphisms in gauge-equivariantK-theory groups [15, Theorem 3.18]

Ki±1G (X × R, Y × R) ' Ki

G(X,Y )

can be composed with ϕF1

, the Thom morphism for F 1, giving a morphism

(16) ϕF : KiG(X) → Ki

G(F ) , i = 0, 1.

This morphism is the Thom morphism for K1.Let pX : X → B and pY : Y → B be two compact G-fiber bundles. Let

πE : E → X and πF : F → Y be two complex G–equivariant vector bundles.Denote by p1 : X×B Y → X and by p2 : X×B Y → Y the projections onto the twofactors and define E � F := p∗1E ⊗ p∗2F . The G–equivariant vector bundle E � Fwill be called the external tensor product of E and F over B. It is a vector bundleover X ×B Y . Then the formula

K0G(X) ⊗K0

G(Y ) 3 [E] ⊗ [F ] → [E] � [F ] := [E � F ] ∈ K0G(X ×B Y )

defines a product K0G(X) ⊗K0

G(Y ) → K0G(X ×B Y ).

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18 VICTOR NISTOR AND EVGENIJ TROITSKY

In particular, consider two complex G–equivariant vector bundles πE : E → Xand πF : F → X . Then E ⊕ F = E ×X F and we obtain a product

KiG(E) ⊗Kj

G(F ) 3 [E] ⊗ [F ] → [E] � [F ] := [E � F ] → K i+jG (E ⊕ F ).

Using also periodicity, we obtain the product

(17) � : KiG(E) ⊗Kj

G(F ) → Ki+jG (E ⊕ F ).

This product is again seen to be given by the tensor product of the (lifted) complexes(when representing K-theory classes by complexes) as in the classical case.

The external product � behaves well with respect to the “Thom construction,”in the following sense. Let F 1 and F 2 be two complex bundles over X, and s1, s2two corresponding sections of these bundles. Then

(18) Λ(F 1 ⊕ F 2, s1 ⊗ 1 + 1 ⊗ s2) = Λ(F 1, s1) � Λ(F 2, s2).

In particular, if X is compact, we obtain

(19) λE � λF = λE⊕F .

We shall write s1 + s2 = s1 ⊗ 1 + 1 ⊗ s2, for simplicity.The following theorem states that the Thom class is multiplicative with respect

to direct sums of vector bundles (see also [5]).

Theorem 4.4. Let E,F → X be two G–equivariant vector bundles, and regardE⊕F → E as the G–equivariant vector bundle π∗

E(F ) over E. Then ϕπ∗E(F ) ◦ϕE =

ϕE⊕F .

The above theorem amounts to the commutativity of the diagram

(20) K∗G(X)

ϕE

//

ϕE⊕F

&&LLLLLLLLLLK∗

G(E)

ϕπ∗E

(F )xxrrrrrrrrrr

K∗G(E ⊕ F )

Proof. Let F 1 := π∗E(F ) = E ⊕ F , regarded as a vector bundle over E. Consider

the projections

πE : E → X, πF : F → X, πE⊕F : E ⊕ F → X, t = πF 1 : E ⊕ F → E.

Let x ∈ K0G(X). Then ϕE(x) = π∗

E(x)⊗Λ(π∗E(E), sE). Now we use that t∗π∗

E(x) =π∗E⊕F (x) and t∗Λ(π∗

E(E), sE) = Λ(π∗E⊕F (E), sE ◦ t). Since sE ◦ t + sF 1 = sE⊕F ,

Equations (18) and (19) then give

Λ(π∗E⊕F (E ⊕ F ), sE⊕F ) = Λ(π∗

E⊕F (E), sE ◦ t) ⊗ Λ(π∗E⊕F (F ), sF 1).

Putting together the above calculations we obtain

ϕF1

ϕE(x) = t∗(ϕE(x)) ⊗ Λ(t∗(F 1), sF 1)

= t∗π∗E(x) ⊗ t∗(Λ(π∗

E(E), sE)) ⊗ Λ(t∗(F 1), sF 1)

= π∗E⊕F (x) ⊗ Λ(π∗

E⊕FE, sE ◦ t) ⊗ Λ(π∗E⊕FF, sF 1)

= π∗E⊕F (x) ⊗ Λ(π∗

E⊕F (E ⊕ F ), sE⊕F ) = ϕE⊕F1 (x).

The proof is now complete. �

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THOM ISOMORPHISM 19

We are now ready to formulate and prove the Thom isomorphism in the settingof gauge-equivariant vector bundles.

Theorem 4.5. Let X → B be a G-bundle and F → X a complex G–equivariantvector bundle, then ϕF : Ki

G(X) → KiG(F ) is an isomorphism.

Proof. Assume first that F is a trivial bundle, that is, that F = X ×B V , whereV → B is a complex, finite-dimensional G–equivariant vector bundle. We continueto assume that B is compact.

Let us denote by C := B ×C the 1-dimensional G-bundle with the trivial actionof G on C. Also, let us denote by P (V ⊕ C) the projective space associated toV ⊕ C. As a topological space, P (V ⊕ C) identifies with the fiberwise one-pointcompactification of V . The embeddings V ⊂ P (V⊕C) and V×BX ⊂ P (V⊕C)×BXthen gives rise to the following natural morphism (Equation (6))

j : K0G(V) → K0

G(P (V ⊕ C)), j : K0G(V ×B X) → K0

G(P (V ⊕ C) ×B X).

Let X be compact and let x ∈ K0G(P (V ⊕ C) ×B X) be arbitrary. The fibers

of the projectivization P (V ⊕ C) are complex manifolds, so we can consider theanalytical index of the correspondent family of Dolbeault operators over P (V ⊕ 1)with coefficients in x (cf. [3, page 123]). This index is an element of K0

G(X) by theresults of [15]. Taking the composition with j (cf. [3, page 122-123]) we get a familyof mappings αX : K0

G(V ×B X) → K0G(X), having the following properties:

(i) αX is functorial with the respect to G–equivariant morphisms;(ii) αX is a morphism of K0

G(X)-modules;

(iii) αB(λV ) = 1 ∈ K0G(B).

Let X+ := X ∪ B be the fiberwise one-point compactification of X . The com-mutative diagram

0 → K0G(V ×B X) // K0

G(V ×B X+) //

αX+

��

K0G(V)

αB

��0 → K0

G(X) // K0G(X+) // K0

G(B)

with exact lines allows us to define αX for X non-compact.Let x ∈ K0

G(X), then by (ii)

(21) αX (λV x) = αX (λV)x = x, αϕ = Id .

Let q := πF : F = V ×B X → X , p : X ×B V → X , q1 : V ×B X ×B V → V(onto the first entry), q2 : X ×B V → B, and by y ∈ K0

G(X ×B V) we denote theelement, obtained from y under the mapping X ×B V → V ×B X , (x, v) 7→ (−v, x)(such that V ×B X ×B V → V ×B X ×B V , (u, x, v) 7→ (−v, x, u) is homotopic tothe identity).

Let y ∈ K0G(V ×B X), then once again by (i), (ii) and then by (iii)

(22) ϕ(αX (y)) = π∗EαX(y) ⊗ q∗λV = αX×BV(p∗1y) ⊗ q∗λV = αX×BV(p∗1y ⊗ q∗λV )

= αX×BV(y � λV ) = αX×BV(λV � y) = αX×BV(q∗1λV ⊗ y)

= αX×BV(q∗1λV ) ⊗ y = q∗2αB(λV ) ⊗ y = q∗2(1) ⊗ y = y ∈ K0G(X ×B V),

We obtain that ϕ ◦ αX is an isomorphism. Since αX ◦ ϕ = Id, αX is the two-sidesinverse of ϕ and the automorphism ϕ ◦ αX is the identity.

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20 VICTOR NISTOR AND EVGENIJ TROITSKY

The proof for a general (complex) G–equivariant vector bundle F → X can bedone as in [3, p. 124]. However, we found it more convenient to use the followingargument. Embed first F into a trivial bundle E = V ×B X . Let ϕ1 and ϕ2 be theThom maps associated to the bundles E → F and E → X . Then by Theorem 4.4the diagram

K0G(F )

ϕ1

// K0G(E)

K0G(X)

ϕF

ddIIIIIIIII ϕ2

::uuuuuuuuu

is commutative, while ϕ2 is an isomorphism by the first part of the proof. ThereforeϕF is injective. The same argument show that ϕ1 is injective. But ϕ1 must also besurjective, because ϕ2 is an isomorphism. Thus, ϕ1 is an isomorphism, and henceϕ2 is an isomorphism too. �

5. Gysin maps

We now discuss a few constructions related to the Thom isomorphism, which willbe necessary for the definition of the topological index. The most important oneis the Gysin map. For several of the constructions below, the setting of G-spacesand even G-bundles is too general, and we shall have to consider longitudinallysmooth G–fiber bundles πX : X → B. The main reason why we need longitudinallysmooth bundles to define the Gysin map is the same as in the definition of theGysin map for embeddings of smooth manifolds. We shall denote by TvertX thevertical tangent bundle to the fibers of X → B. All tangent bundles below will bevertical tangent bundles.

Let X and Y be longitudinally smooth G-fiber bundles, i : X → Y be anequivariant fiberwise embedding, and pT : TvertX → X be the vertical tangentbundle to X. Assume Y is equipped with a G-invariant Riemannian metric and letpN : Nvert → X be the fiberwise normal bundle to the image of i

Let us choose a function ε : X → (0,∞) such that the map of Nvert to itself

n 7→ εn

1 + |n|

is G-equivariant and defines a G-diffeomorphism Φ : Nvert → W onto a bundle ofopen tubular neighborhoods W ⊃ X in Y.

Let (N ⊕N)vert := Nvert ⊕Nvert. The embedding i : X → Y can be written asa composition of two fiberwise embeddings i1 : X → W and i2 : W → Y . Passingto differentials we obtain

TvertXdi1−→ TvertW

di2−→ TvertY and dΦ : TvertN → TvertW,

where we use the simplified notation TvertN = TvertNvert.

Lemma 5.1. (cf. [10, page 112]) The manifold TvertN can be identified withp∗T (N ⊕N)vert with the help of a G-equivariant diffeomorphism ψ such that makes

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THOM ISOMORPHISM 21

the following diagram commutative

p∗T (N ⊕N)vert

��

TvertNψ

oo

��TvertX

pT

&&MMMMMMMMMMMNvert

pN

{{wwww

wwww

w

X.

Proof. The vertical tangent bundle TvertN → Nvert and the vector bundle

p∗N (TvertX) ⊕ p∗N (Nvert) → Nvert

are isomorphic as G–equivariant vector bundles over Nvert.Indeed, a point of the total space TvertN is a pair of the form (n1, t + n2),

where both vectors are from the fiber over the point x ∈ X. Similarly, we representelements p∗T (N ⊕ N)vert as pairs of the form (t, n1 + n2). Let us define ψ by theequality ψ(n1, t+ n2) = (t, n1 + n2). �

With the help of the relation i · (n1, n2) = (−n2, n1), we can equip

p∗T (N ⊕N)vert = p∗T (Nvert) ⊕ p∗T (Nvert)

with a structure of a complex manifold. Then we can consider the Thom homo-morphism

ϕ : K0G(TvertX) → K0

G(p∗T (N ⊕N)vert).

Since TvertW is an open G-stable subset of TvertY and di2 : TvertW → TvertYis a fiberwise embedding, by Equation 6, there is the homomorphism (di2)∗ :K0

G(TvertW ) → K0G(TvertY ).

Definition 5.2. Let i : X → Y be an equivariant embedding of G-bundles. TheGysin homomorphism is the mapping

i! : K0G(TvertX) → K0

G(TvertY ), i! = (di2)∗ ◦ (dΦ−1)∗ ◦ ψ∗ ◦ ϕ.

In other words, it is obtained by passage to K-groups in the upper part of thediagram

p∗T (N ⊕N)vert

qT

��

TvertNψoo dΦ //

��

TvertWdi2 //

��

TvertY

��

TvertXpT

&&MMMMMMMMMMMNvert

pN

{{wwww

wwww

%%KKKKKKKKKK

Xi1 // W

i2 // Y.

Another choice of metric and neighborhood W induces the homotopic map and(by the item 3 of Theorem 5.3 below) the same homomorphism.

Theorem 5.3 (Properties of Gysin homomorphism). Let i : X → Y be a G-em-bedding.

(i) i! is a homomorphism of K0G(B)-modules.

(ii) Let i : X → Y and j : Y → Z be two fiberwise G-embeddings, then (j ◦ i)! =j! ◦ i!.

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22 VICTOR NISTOR AND EVGENIJ TROITSKY

(iii) Let fiberwise embeddings i1 : X → Y and i2 : X → Y be G-homotopic in theclass of embeddings. Then (i1)! = (i2)!.

(iv) Let i! : X → Y be a fiberwise G-diffeomorphism, then i! = (di−1)∗.(v) A fiberwise embedding i : X → Y can be represented as a compositions of

embeddings X in Nvert (as the zero section s0 : x → N) and Nvert → Y byi2 ◦ Φ : Nvert → Y. Then i! = (i2 ◦ Φ)!(s0)!.

(vi) Consider the complex bundle p∗T (Nvert ⊗ C) over TvertX. Let us form thecomplex Λ(p∗T (Nvert ⊗ C), 0) :

0 → Λ0(p∗T (Nvert ⊗ C))0

−→ . . .0

−→ Λk(p∗T (Nvert ⊗ C)) → 0

with noncompact support. If a ∈ K0G(TvertX), then the complex

a⊗ Λ(p∗T (Nvert ⊗ C), 0)

has compact support and defines an element of K0G(TvertX). Then

(di)∗i!(a) = a · Λ(p∗T (Nvert ⊗ C), 0),

where di is the differential of the embedding i.(vii) i!(x(di)

∗y) = i!(x) · y, where x ∈ K0G(TvertX) and y ∈ K0

G(TvertY ).

Proof. (i) This follows from the definition of i!.(ii) To simplify the argument, let us identify the tubular neighborhood with the

normal bundle. Then (j ◦ i)! is the composition

K0G(TvertX)

ϕ−→ K0

G(TvertN ⊕ TvertN′′vert) → K0

G(TvertZ),

where N ′vert is the fiberwise normal bundle of Y in Z, N ′′

vert = N ′vert|X , and for

the sum of tangent bundles to the vertical normal bundles TvertN ⊕ TvertN′′vert is

considered in the same way as on page 21, that is, as a complex bundle over TvertX .On the other hand, j! ◦ i! represents the composition

K0G(TvertX)

ϕ−→ K0

G(TvertN) → K0G(TvertY )

ϕ−→ K0

G(TvertN′vert) → K0

G(TvertZ).

By the properties of ϕ, the following diagram is commutative

K0G(TvertX)

ϕ //

ϕ

))SSSSSSSSSSSSSSK0

G(TvertN) //

ϕ

��

K0G(TvertY )

ϕ

��K0

G(TvertN ⊕ TvertN′vert)

ϕ //

��

K0G(TvertN

′vert)

��K0

G(TvertZ) K0G(TvertZ).

This completes the proof of (ii).(iii) The morphism qT depends only on the homotopy class of the embeddings

used to define it. The assertion thus follows from the homotopy invariance of K-theory.

(iv) In this case N = X, W = Y, Φ = i, i2 = IdY , and the formula is obvious.(v) This follows from (ii).(vi) By definition,

(di)∗ ◦ i! = (di1)∗ ◦ (di2)

∗ ◦ (di2)∗ ◦ (dϕ−1)∗ ◦ ψ∗ ◦ ϕ∗ = (ψ ◦ dΦ−1 ◦ di1)∗ ◦ ϕ,

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THOM ISOMORPHISM 23

where i1 : X → W, i2 : W → Y. Let (n1, t + n2) ∈ TvertN = p∗N(TvertX) ⊕p∗N(Nvert), where n1 is the shift under the exponential mapping and t + n2 is avertical tangent vector to W. If dΦ(n1, t + n2) is in TvertX , then n1 = n2 = 0.Hence,

dΦ−1di1(t) = (0, t+ 0), ψ ◦ dΦ−1 ◦ di1(t) = (t, 0 + 0).

Therefore, ψ ◦ dΦ−1 ◦ di1 : TvertX → p∗T (Nvert ⊕ Nvert) is the embedding of thezero section. Since ϕ(a) = a · Λ(q∗T p

∗T (Nvert ⊗ C), sp∗

T(Nvert⊗C)), it follows that

(di)∗ ◦ i!(a) = a · Λ(p∗T (Nvert ⊗ C), 0).(vii) The mapping di1 ◦ qT ◦ ψ ◦ dΦ−1 : TvertW → TvertW is homotopic to the

identical mapping. Hence,

i!(x · (di)∗y) = (di2)∗(dΦ−1)∗ψ∗ϕ(x · (di)∗y) =

= (di2)∗(dΦ−1)∗ψ∗[(q∗T (x)λp∗

T(Nvert⊗C))(q

∗T (di)∗y)] =

= (di2)∗[(dΦ−1)∗ψ∗(q∗T (x)λp∗

T(Nvert⊗C)) (dΦ−1)∗ψ∗q∗T (di1)

︸ ︷︷ ︸Id

(di2)∗y] =

= [(di2)∗(dΦ−1)∗ψ∗(q∗T (x)λp∗

T(Nvert⊗C))] [(di2)∗(di2)

∗y] = i!(x) · y.

The proof is now complete. �

We shall need also the following properties of the Gysin map. If X = B, thetrivial longitudinally smooth G-bundle, we shall identify TvertX = B and TvertV =V ⊗ C for a real bundle V → B.

Theorem 5.4. Suppose that V → B is a G-equivariant real vector bundle and thatX = B. Then the mapping

i! : K0G(B) = K0

G(TXvert) → K0G(TvertV) = K0

G(V ⊗ C)

coincides with the Thom homomorphism ϕV⊗C.

Proof. The assertion follows from the definition of i!. More precisely, let X =B ↪→ V , N = V be the zero section embedding. In the definition of the Thomisomorphism, W can be chosen to be equal to the bundle D1 of interiors of theballs of radius 1 in V with respect to an invariant metric. In this case, the diagramfrom the definition of the Gysin homomorphism 5.2 takes the following form

V ⊗ C

��

Ψ // V ⊗ CdΦ //

��

D1 ⊗ Cdi2 //

��

V ⊗ C

��

TvertX = B

&&NNNNNNNNNNNV

zztttttt

tttt

%%JJJJJJJJJJJ

X = Bi1 // D1

i2 // V .

In our case Ψ = Id and di2 ◦ dΦ is homotopic to Id, since this map has the formv ⊗ z 7→ (v ⊗ z)/(1 + |v ⊗ z|). Hence, i! = ϕ. �

Theorem 5.5. Suppose that V ′ and V ′′ are G-equivariant R-vector bundles over Band that i : X → V ′ is an embedding. Let k : X → V ′ ⊕ V ′′, k(x) = i(x) + 0. Let ϕbe the Thom homomorphism of the complex bundle

Tvert(V′ ⊕ V ′′) = V ′ ⊗ C ⊕ V ′′ ⊗ C −→ TvertV

′ = V ′ ⊗ C.

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24 VICTOR NISTOR AND EVGENIJ TROITSKY

Then the following diagram is commutative

K0G(TvertV ′)

ϕ

��

K0G(TvertX)

i!

66lllllllllllll

k! ((RRRRRRRRRRRRR

K0G(Tvert(V ′ ⊕ V ′′)).

Proof. To prove the statement, let us consider the embedding i : X → V ′ anddenote, as before, by Nvert the fiberwise normal bundle and by W the tubularneighborhood appearing in the definition of the Thom isomorphism associated to i.Then Nvert⊕V ′′ is a fiberwise normal bundle for the embedding k with the tubularneighborhood W ⊕D(V ′′), where D(V ′′) is a ball bundle. If a ∈ K0

G(TvertX), then

k!(a) = (di2 ⊕ 1)∗ ◦ (dΦ−1 ⊕ 1)∗ ◦ (ψ ⊕ 1)∗ ◦ ϕN⊕N⊕V′′⊕V′′

(a).

We have ϕN⊕N⊕V′′⊕V′′

= ϕN⊕N ◦ϕV′′⊕V′′

, by Theorem 4.4. Since a = a ·C, whereC is the trivial line bundle, we obtain

(23) k!(a) = (di2)∗ ◦ (dΦ−1)∗ ◦ Ψ∗ ◦ ϕNvert⊕Nvert(a) ◦ ϕV′′⊕V′′

(C)

= i!(a) · λT (V′⊕V′′)vert= ϕ(i!(a)).

The proof is now complete. �

6. The topological index

We begin with a “fibered Mostow-Palais theorem” that will be useful in definingthe index.

Theorem 6.1. Let πX : X → B be a compact G-fiber bundle. Then there existsa real G–equivariant vector bundle V → B and a fiberwise smooth G-embeddingX → V. After averaging one can assume that the action of G on V is orthogonal.

Proof. Fix b ∈ B and let U be an equivariant trivialization neighborhood of b forboth X and G. By the Mostow-Palais theorem, there exists a representation ofGb on a finite dimensional vector space Vb and a smooth Gb-equivariant embeddingib : Xb → Vb. This defines an embedding

(24) ψ : π−1X (Ub) ' Ub ×Xb → Ub × Vb

which is G-equivariant in an obvious sense.We can cover B with finitely many open sets Ubj

, as above, corresponding to thepoints bj , j = 1, . . . , N . Denote by Vj the corresponding representations and by ψjthe corresponding embeddings, as in Equation (24). Let W := ⊕Vj . Also, Let φjbe a partition of unity subordinated to the covering by Uj = Ubj

. We define then

Ψ := ⊕(φj ◦ πX )ψj : X → B ×W,

which is a G-equivariant embedding of X into the trivial G–equivariant vector bun-dle V := B ×W , as desired. �

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THOM ISOMORPHISM 25

Let us now turn to the definition of the topological index. Let X → B bea compact, longitudinally smooth G-bundle. From Theorem 6.1 it follows thatthere exists an G–equivariant real vector bundle V → B and a fiberwise smoothG-equivariant embedding i : X → V . We can assume that V is endowed with anorthogonal metric and that G preserves this metric. Thus, the Gysin homomorphism

i! : K0G(TvertX) → K0

G(TvertV) = K0G(V ⊗ C)

is defined (see Section 4). Since TvertV = V ⊗ C is a complex vector bundle, wehave the following Thom isomorphism (see Section 4):

ϕ : K0G(B)

∼−→ K0

G(TvertV).

Definition 6.2. The topological index is by definition the morphism:

t-indXG : K0G(TvertX) → K0

G(B), t-indXG := ϕ−1 ◦ i!.

The topological index satisfies the following properties.

Theorem 6.3. Let X → B be a longitudinally smooth bundle and

t-indXG : K0G(TvertX) → K0

G(B)

be its associated topological index. Then

(i) t-indXG does not depend on the choice of the G–equivariant vector bundle Vand on the embedding i : X → V.

(ii) t-indXG is a K0G(B)-homomorphism.

(iii) If X = B, then the map

t-indXG : K0G(B) = K0

G(TvertX) → K0G(B)

coincides with IdK0G(B).

(iv) Suppose X and Y are compact longitudinally smooth G-bundles, i : X → Yis a fiberwise G-embedding. Then the diagram

K0G(TvertX)

i! //

t-indXG &&MMMMMMMMMM

K0G(TvertY )

t-indYGxxqqqqqqqqqq

K0G(B).

commutes.

Proof. To prove (i), let us consider two embeddings

i1 : X → V ′, i2 : X → V ′′

into G–equivariant vector bundles. Denote by j = i1 + i2 the induced embeddingj : X → V ′ ⊕ V ′′. It is sufficient to show that i1 and j define the same topologicalindex. Let us define a homotopy of G-embeddings by the formula

js(x) = i1(x) + s · i2(x) : X → V ′ ⊕ V ′′, 0 ≤ s ≤ 1.

Then, by Theorems 5.3(iii) and 5.5, the indices for j and j0 coincide. Let us shownow that j0 = i1 + 0 and i1 define the same topological indexes. For this purpose

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26 VICTOR NISTOR AND EVGENIJ TROITSKY

consider the diagram

K0G(TvertX)

(i1)!

wwooooooooooo(j0)!

((RRRRRRRRRRRRR

K0G(TvertV ′)

ϕ2 // K0G(Tvert(V ′ ⊕ V ′′)),

K0G(B)

ϕ1

ggOOOOOOOOOOO ϕ3

66llllllllllllll

where ϕi are the corresponding Thom homomorphisms. The upper triangle iscommutative by Theorem 5.4.2, and the lower is commutative by Theorem 4.4.Hence ϕ−1

1 ◦ (i1)! = ϕ−13 ◦ (j0)1 as desired.

(ii) follows from 4.2 and 5.3(i).Property (iii) follows from the definition of the index and from 5.4.To prove (iv), let us consider the diagram

Xi //

j◦i AAA

AAAA

A Y

j~~}}}}

}}}

V .

We now use 5.3(ii). This gives the commutative diagram

K0G(TvertX)

i! //

(j◦i)! ''OOOOOOOOOOOK0

G(TvertY )

j!wwooooooooooo

K0G(TvertV)

or

K0G(TvertX)

i! //

(j◦i)!

''OOOOOOOOOOO

t-indXG

��>>>

>>>>

>>>>

>>>>

>>>>

K0G(TvertY )

j!

wwooooooooooo

t-indYG

������

����

����

����

���

K0G(TvertV)

K0G(B).

ϕ

OO

This completes the proof. �

We now investigate the behavior of the topological index with respect to fiberproducts of bundles of compact groups.

Theorem 6.4. Let π : P → X be a principal right H-bundle with a left action ofG commuting with H. Suppose F is a longitudinally smooth (G × H)-bundle. Letus denote by Y the space P ×H F . Let j : X ′ → X and k : F ′ → F be fiberwise G-and (G × H)-embeddings, respectively. Let π′ : P ′ → X ′ be the principal H-bundleinduced by j on X ′. Assume that Y ′ := P ′ ×H F ′. The embeddings j and k induce

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THOM ISOMORPHISM 27

G-embedding j ∗ k : Y ′ → Y . Then the diagram

K0G(TvertX) ⊗K0

G(B) K

0G×H(TvertF )

γ // K0G(TvertY )

K0G(TvertX

′) ⊗K0G(B) K

0G×H(TvertF

′)γ //

j!⊗k!

OO

K0G(TvertY

′)

(j∗k)!

OO

is commutative.

Let us remark that there in the statement of this theorem there is no compactnessassumption on X,X ′, F, and F ′, since there is no compactness assumption in thedefinition of the Gysin homomorphism. This is unlike in the definition of thetopological index where we start with a compact G-bundle X → B.

Proof. Let us use the definition of γ:

K0G(TvertX) ⊗K0

G×H(TvertF ) //

1

K0G(TvertX) ⊗K0

G×H(P × TvertF )∼= //

2

K0G(TvertX

′) ⊗K0G×H(TvertF

′) //

j!⊗k!

OO

K0G(TvertX

′) ⊗K0G×H(P ′ × TvertF

′)

ε

OO

∼= //

∼= K0G(TvertX) ⊗K0

G(P ×H TvertF ) →

3

K0G(π∗

1TvertX) ⊗K0G(P ×H TvertF ) →

∼= K0G(TX ′) ⊗K0

G(P ′ ×H TvertF′) →

β

OO

K0G((π′

1)∗TvertX

′) ⊗K0G(P ′ ×H TvertF

′) →

α

OO

(25)

→ K0G((π∗

1TvertX) × (P ×H TvertF )) →

4→ K0

G((π′1)

∗TvertX′ × (P ′ ×H TvertF

′)) →

→ K0G(π∗

1TvertX ⊕ (P ×H TvertF )) = K0G(TvertY )

4 ↑ (j ∗ k)!→ K0

G((π′1)

∗TvertX′ ⊕ (P ′ ×H TvertF

′)) = K0G(TY ′

vert),

where projections π1 : Y = P ×H F → X and π′1 : Y ′ = P ×H F ′ → X ′ are defined

as above. Here we use the isomorphism K0G×H(P ×W ) ∼= K0

G(P ×H W ) for a freeH-bundle P (see Theorem 2.6). Let us remind the diagram, which was used for thedefinition of the Gysin homomorphism of an embedding j : X ′ → X :

p∗T (NX′ ⊕NX′)vert

qX′

T

��

TvertNX′

ψoo dΦX′ //

��

TvertWX′

dj2 //

��

TvertX

��

TvertX′

pT

''OOOOOOOOOOOOONX′,vert

pNX′,vert

zzuuuuuuuuuΦX′

&&MMMMMMMMMM

X ′j1 // WX′

j2 // X.

From the similar diagrams for k! and (j ∗k)! and the explicit form of these maps, it

follows that the square 4 in (25) is commutative if, and only if, α has the following

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28 VICTOR NISTOR AND EVGENIJ TROITSKY

form:

α(σ ⊗ ρ) = (π∗1){(dj2)∗ (dΦ−1

X′ )∗ ψ∗X′

}◦ ϕS(σ)⊗

⊗(π∗j2 ×H dk2)∗

((π∗ΦX′ ×H dΦF ′)−1

)∗(1 ×H ψF ′)∗ ϕR(ρ),

where S and T are bundles of the form

π∗1

((pX

T )∗{NX′ ⊕NX′})

π∗NX′ ×H (pF′

T )∗ (NF ′ ⊕NF ′)

S : ↓ (π′1)

∗qX′

T R : ↓ (π′)∗(pNX′ ) ×H qF′

T

(π′1)

∗ (TvertX′), π∗X ′ ×H TvertF

′ = P ′ ×H TvertF′.

Hence the square 3 in (25) is commutative iff the homomorphism β has the form

β(τ ⊗ ρ) = j!(τ)⊗

⊗(π∗j2 ×H dk2)∗

((π∗ΦX′ ×H dΦF ′)−1

)∗(1 ×H ψF ′)∗ ϕR(ρ),

τ ∈ K0G(TX ′), ρ ∈ K0

G(P ′ ×H TF ′).

In turn, the square 2 in (25) is commutative iff the homomorphism ε has the form

ε(τ ⊗ δ) = j!(τ)⊗

⊗(π∗j2 ×H dk2)∗

((π∗ΦX′ ×H dΦF ′)−1

)∗(1 ×H ψF ′)∗ ϕ

eRC(δ),

τ ∈ K0G(TX ′), δ ∈ K0

G×H(P ′ × TF ′),

where R is the following bundle:

π∗NX′ × (pF′

T )∗ (NF ′ ⊕NF ′)

R : ↓ (π′)∗(pN ) × qF′

T

P ′ × TF ′.

Suppose δ = [C]⊗ω, where [C] ∈ K0G×H(P ′), C is the one-dimensional trivial bundle

and ω ∈ K0G×H(TF ′). Then

ε(τ ⊗ δ) = j!(τ) ⊗{π∗(j2)∗(Φ

−1X′ )

∗[C]⊗k!(ω)}

=

= j!(τ) ⊗{[C]⊗k!(ω)

}.

Since the map K0G×H(TF ) → K0

G×H(P ×TF ) (as well as the lower line in (25)) has

the form ω 7→ [C]⊗ω, we have proved the commutativity of 1 in (25). �

From this theorem we obtain the following corollary.

Corollary 6.5. Let M be a compact smooth H-manifold, let H = B ×H, and letP be a principal longitudinally smooth H-bundle over X carrying also an action ofG commuting with the action of H. Also, let X → B be a compact longitudinallysmooth G-bundle. Let Y := P ×H M → X be associated longitudinally smoothG-bundle. Taking F = B × M , we define TMY := TFY . Then TMY is a G-invariant real subbundle of TvertY and TMY = P ×H TM . Let j : X ′ → X bea fiberwise G–equivariant embedding and let k : M ′ → M be an H-embedding.Denote by π′ : P ′ → X ′ the principal H-bundle induced by j on X ′ and assume

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THOM ISOMORPHISM 29

that Y ′ := P ′ ×HM ′. The embeddings j and k induce G-embedding j ∗ k : Y ′ → Y .Then the diagram

K0G(TvertX) ⊗K0

H(TM)γ // K0

G(TvertY )

K0G(TvertX

′) ⊗K0H(TM ′)

γ //

j!⊗k!

OO

K0G(TvertY

′)

(j∗k)!

OO

is commutative.

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Department of Mathematics, Pennsylvania State University

E-mail address: [email protected]

URL: http://www.math.psu.edu/nistor

Dept. of Mech. and Math., Moscow State University, 119992 Moscow, Russia

E-mail address: [email protected]

URL: http://mech.math.msu.su/~troitsky